Problem: Simplify and expand the following expression: $ \dfrac{2r}{r + 6}+\dfrac{r + 5}{r + 2} $
Solution: In order to add expressions, they must have a common denominator. Get both fractions over a common denominator of $(r + 6)(r + 2)$ Multiply the first term by $\dfrac{r + 2}{r + 2}$ $ \begin{align*} \dfrac{2r}{r + 6} \times \dfrac{r + 2}{r + 2} & = \dfrac{(2r)(r + 2)}{(r + 6)(r + 2)} \\ & = \dfrac{2r^2 + 4r}{(r + 6)(r + 2)}\end{align*} $ Multiply the second term by $\dfrac{r + 6}{r + 6}$ $ \begin{align*} \dfrac{r + 5}{r + 2} \times \dfrac{r + 6}{r + 6} & = \dfrac{(r + 5)(r + 6)}{(r + 2)(r + 6)} \\ & = \dfrac{r^2 + 11r + 30}{(r + 2)(r + 6)}\end{align*} $ Now we have: $ = \dfrac{2r^2 + 4r}{(r + 6)(r + 2)} + \dfrac{r^2 + 11r + 30}{(r + 2)(r + 6)} $ Now both terms have a common denominator we can simply add the numerators: $ = \dfrac{2r^2 + 4r + r^2 + 11r + 30}{(r + 6)(r + 2)} $ $ = \dfrac{3r^2 + 15r + 30}{(r + 6)(r + 2)}$ Expand the denominator: $ = \dfrac{3r^2 + 15r + 30}{r^2 + 8r + 12}$